Quantum Probability & the Measurement ProblemQuantum physics is defined mathematically by the Schroedinger equation, which depicts the probability of a particle being found at a certain point. This probability is fundamental to the system, not merely a result of ignorance. Once a measurement is made, however, you have a definite result.
The measurement problem is that the theory doesn't completely explain how the act of measurement actually causes this change. Attempts to solve the problem have lead to some intriguing theories.
For example, the more precisely you measure the momentum of a particle the less precise your measurement of its position. Again, in Heisenberg's interpretation this wasn't just a measurement error or technological limitation, but an actual physical limit.
Quantum Entanglement & NonlocalityIn quantum theory, certain physical systems can become "entangled," meaning that their states are directly related to the state of another object somewhere else. When one object is measured, and the Schroedinger wavefunction collapses into a single state, the other object collapses into its corresponding state ... no matter how far away the objects are (i.e. nonlocality).
Einstein, who called these influences "spooky action at a distance," illuminated this concept with his EPR Paradox.
In 1929, Edwin Hubble discovered a redshift in distant galaxies, indicating that they were receding from Earth. Cosmic background microwave radiation, discovered in 1965, supported Lemaitre's theory.
Dark Matter & Dark EnergyAcross astronomical distances, the only significant fundamental force of physics is gravity. Astronomers find that their calculations & observations don't quite match up, though.
Other work indicates that there might exist a dark energy, as well.
Quantum ConsciousnessIn attempts to solve the measurement problem in quantum physics (see above), physicists frequently run into the problem of consciousness. Though most physicists try to sidestep the issue, it seems that there is a link between the conscious choice of experiment and the outcome of the experiment.
Some physicists, most notably Roger Penrose, believe that current physics cannot explain consciousness, and that consciousness itself has a link to the strange quantum realm.
The controversial Anthropic Principle states that the universe can only exist such that carbon-based life can arise.
The Anthropic Principle, while intriguing, is more a philosophical theory than a physical one. Still, the Anthropic Principle poses an intriguing intellectual puzzle.